(i) State the condition on the real numbers b and c under which the quadratic expression x2+bx+c has no real roots.
(ii) Give a reason why, for a proper rational function whose denominator is an irreducible quadratic x2+bx+c, the most general numerator must be taken to be the linear expression Ax+B even if a simpler form might appear to work for a specific example.
(iii) Let
I=∫x2+4x+53x+7dx.
Write the denominator in completed–square form and hence indicate, without actually carrying out the integration, the two elementary integrals into which I can be split.